Signal-Pressure Curves of Cascaded Four-Wave Mixing in Gas-Filled Capillary by fs Pulses

Commun. Theor. Phys.(Beijing, China)43(2005)pp 526-528C International Academic PublisheVol. 43. No. 3. March 15. 2005Signal-Pressure Curves of Cascaded Four-Wave Mixing in Gas-Filled Capillary by fsPulsesChen Bao-Zhen and HUANG Zu-QiaThe Key Laboratory of Beam Technology and Materials Modification of Ministry of Education, Institute of Low EnergNuclear Physics, Beijing Normal University, Beijing 100875, China(Received June 17, 2004; Revised August 20, 2004)Abstract The theoretical framework for the cascaded four waves mixing(CFWM) in gas-filled capillary by fs pulsesis constructed. Based on the theoretical framework, the signal-pressure curves(SPC)of the CFWM in gas-filled capillaryby fs pulses are calculated. With a comparison between the theoretical and experimental SPC we have discussed theinfluence of the walk-off and phase modulation on the SPC. At the same time, we have discussed the possible origin ofthe first three peaks of the SPcPACS numbers: 42.50.Gy, 42.65.Ky, 42.81.QbKey words: gas-filled capillary, fs pulses, cascaded four-wave mixing, signal-pressure curvesThe experiments in Ref. I showed that the efficiency w2= 2w, w3= 3w, W4= 4w). According to Ref. 2, theof the tetramerous frequency generated by the cascaded electric field amplitude of the tetramerous, triple, pumpfour-wave mixing(CFWM)in gas-filled capillary is higher and idler frequencies in the capillary, E4, E3, E2, E1,canthan 1%. The experiments have demonstrated an im- be expanded with the guided modes of the capillary aspressive progress in the area of generating coherent UV follow200 nm) signal of ultra-short pulses by the CFWM in Egas-filled capillary. Obviously, the results of the experi∑[An(z,1)3/(ments stimulated much interest of intensive research on=1,2,3,4nonlinear-optical frequency conversion in gas-filled capil- Here, Sjo(r, o)is the transverse distribution of the pulsecurves of the tetramerous frequency generated by the electric field corresponding to the frequency w, and theCFWM were also given in Ref. [1]. These experimen- eigenmode of the capillary with mode index a, (o is antal curves make it possible to understand in detabbreviation of (n, m), which means the m-th zero pointCFWM in gas-filled capillary by femto-second laserof the n-1 order Bessel function). Ajo(a, t),uo, andfrom theoretical angle. To our knowledge, a theoreticalwin(wi)fit to the curves has not appeared in literature. Suchtheoretical try is given in this letterthe slowly varying amplitude, the eigenvalue and theAccording to the conjecture given in Ref [1], we re- propagation constant of the corresponding eigenmodestrict our investigation to the following CFWM processes respectively. 3 a is the internal radius of the capillaryand the difference frequencies of four-wave mixing(DF- n(w) is the index of refraction for the filled gasFWM):w4=W3+w2-wl and w3= 2w2-w1. Here, w4 is Using a procedure similar to that described in Refs. [4the tetramerous frequency (tF) generated by the CFWM, and 5, the coupled equations for the slowly varying amw2 pump frequency and w idler frequency (wl =w, plitude, Aia(z, t),j=1, 2, 3, 4, can be written as followsOA1o(z,n)=i700140(,m)2+2A2n(2,m2)2A1(,m),A2o0(z,m2)2+2A1o(z,m)2]A2on(2,m)0A3(z,73)A1o0(z,m31)2+|A2o0(z,m32)/2]A3(z,m3)Aigo(expi△32],(3)aA(22=120(,m)2+14an(x,m2)24(2,n)中国煤化工CNMHGA20(2,742)A10(2,741)A3(2,n43)expiThe project supported by National Natural Science Foundation of China under Grant No. 90103025 and Doctoral Program Foundationof Institute of higher education of china under grant No. 20020027006No. 3Signal-Pressure Curves of Cascaded Four-Wave Mixing in Gas-Filled Capillary by fs Pulses527Equations(1)a(4)are, respectively, written in a moving coordinate system defined by the formula z= z, nj=1, 2, 3, 4. To obtain the coupled equations(1)o(4), the following assumptions are used. i) The pump and idlerpulses excite only the lowest mode, that is, a= 00 =(1, 1); ii)According to Ref. [1], the following CFWM processW4=2 Xw3-w2 is omitted; iii)The phase modulations by A3a are omitted. In the coupled equations(1)N(4)A1n(z,t)≡DnA(z,t)=mk+k12,(k,j=1,2,3,4)3丌u23丌u=1~42;n2o3Sa(r, 0)Sa,(r, eSa(,0)Sas(r, e)rdrdeDaDa, da, Da∫2(r,0)321(r, A)rdrdeD=/92(r,0) drdo;△3n=212n0-B1o0-g,=B2oo-B1oo+ B3a,-B4a. The vgj is the group velocity of the pulse with the central frequency w and theparameters have their usual meaningSolving the coupled equations(1)x(4)after Refs. [4] and 6], we obtain the following expressions for the amplitudesin the cfWm and DFfwm casesAlgo(z, m)=Alo(m)expi1(1)), Alo(n)=Algo(0, m1)A2on0(z,m2)=A20(72)expi重2(m2),A20(m2)≡A2on(0,0m2)A3a(2,73)=in anando expli@ 3a(/)/A2(032)Aio(m31)expliT 3o (2)dz'Ao=in40odog,explipAo(74)1/ Aioo(2, n41)A3a(2, 743)A2oo(, 742)exp[ ao(=')dz1(mn)=mo4(mn)2+2/A0(m2)2d2,2(m2)=7am0m)22+2/1410(h2)2dI3n(23)=△3nz+2重2(n2)-更1(n31)-Φ3n(n3),43n(m)=23m/1A20(m2)2+1410(h1)d2,(m)=2/ta0(4(man)2+1n(m)2)dUsing the approximate solution( 8), we have calculated the SPC for the TF signals generated by the CFWM processeswhich are shown in figs. 1 and 2The following parameters are used in the calculations: pump wave length 0. 4 um, idler wave length 0.8 um, capillarylength 80 cm, capillary radius 62.5 um, duration of pump(idler) pulses 25 fs, pump energy in a pulse 25 uJ, idlerenergy in a pulse 30 uJ;x(3)(Ar)=7.3 x 10-18. 7 The following assumption is used in the calculations. The TF signalsgenerated by the CFWM processes are in the EHll modeThe walk-off and phase modulation are omitted in Fig. 1. It is found from Fig. 1 that the peak positions demonstrated by the theoretical SPC are almost the same as that demonstrated by the experimental one. At the same timewe know that the left peak origins the phase-matched CFWM(A(a)4o0oσ-7.37p+0.39=0), the middle peak originat which the DFFWM(w3=2xw2-w1) reaches phase-matched. These results fit the conjecture in Ref. bressurethe quasi-phase-matched CFWM(4a)10.0p+0.71=0)and the right peak responds to theThe phase modulation and walk-off are taken into account in Fig. 2. It is found from Fig. 2 that except the leftpeak(near 40 Torr) the others are not obvious. Therefore, figure 2 shows that the CFWM should not be an only originfor the Tf signals observed in the experimentIt is necessary to find the other processes which can obviously produce the three peaks. In this paper, we take theollowing sum FWM (SFFWM) into account: w4=2X W1 +w2. Afteer a si+o rat ea(7), we can getthe following approximate formula for SFFWM中国煤化工CNMHGA4o(2, n4)=inandoa expli Ao(7A)/A20(042)420(m41)expu ia (2)whereI{(x=△(m2+42(m12)+21(m1)-4(),△1=21m+m0-528CHEn Bao-Zhen and HUANG Zu-QiaVol. 43Using the approximate solution(9), we have calculated the SPC for the TF signals generated by the SFFWm processeswhich are shown in Fig 3From Fig 3, we can see that the theoretical SPC for the Tf signals generated by the SFFWM processes fit theSPC observed in the experimentx102100Pressure(Torr)Pressure(Torr)Fig. 1 Signal-pressure curve for CFWM(without walk-off Fig. 2 Signal-pressure curve for CFWM (with walk-off andand phase modulation)phase modulation)010I三100100Pressure (Torr)Pressure(Torr)Fig.3 Signal-pressure curve for SFFWM (with walk-off and Fig 4 Signal-pressure curve for SFFWM+ CFWM(withoutalk-off and phase modulationSa Csing the approximate solutions(8)and(9), we have calculated the SPC for the TF signals generated by theCo processes and the CFWM processes, which are shown in Fig4ing Figs. 3 and 4, we can see that the SPC for the TF signals generated by the SFFWM processes and theCFWM processes is almost the same as the SPC for the tf signals generated by the SFFWM processes. This is tesay that the contribution to the tf signals from the SFFWM is much larger than that from the CFWmReferences5 G.P. Agrawal, Nonlinear Fiber Optics, Academic Press,[1 L. Misoguti, et al., Phys. Rev. Lett. 87(2001)013601San Diego, USA(1995)e] R.H. Stolen and J E Bjorkholm, IEEE QE 18(1982)1062. [6)N L. Koroteev and A.M. Zheltikov, Appl. Phys.B673 E.A.J. Marcatili and R A. Schmeltzer, Bell Syst. Tech. J43(1964)17837H.J. Lehmeier, W. Leupacher, and A. Pen14 B.Z. Chen and Z.Q. Huang, J. Beijing Normal UniversityCommun. 56(1985)(in Chinese)39(2003)72中国煤化工CNMHG